To solve the problem in the image, we need to find the following values for the data set provided:

1. $\sum x$
2. $\sum x^2$
3. $\left(\sum x\right)^2$

Given Data Set:

$-6, -1, -3, 6, 1, 5, -5, 0, 4$

1. Calculate $\sum x$:

$\sum x = -6 + (-1) + (-3) + 6 + 1 + 5 + (-5) + 0 + 4$

Let's calculate it:

$\sum x = (-6 - 1 - 3 + 6 + 1 + 5 - 5 + 0 + 4) = 1$

2. Calculate $\sum x^2$:

$\sum x^2 = (-6)^2 + (-1)^2 + (-3)^2 + 6^2 + 1^2 + 5^2 + (-5)^2 + 0^2 + 4^2$

Let's calculate each term:

$\sum x^2 = 36 + 1 + 9 + 36 + 1 + 25 + 25 + 0 + 16 = 149$

3. Calculate $\left(\sum x\right)^2$:

$\left(\sum x\right)^2 = (1)^2 = 1$

Results:

• $\sum x = 1$
• $\sum x^2 = 149$
• $\left(\sum x\right)^2 = 1$

Thus, the correct values are:

• $\sum x = 1$
• $\sum x^2 = 149$
• $\left(\sum x\right)^2 = 1$

It looks like the error in the image was because the second calculation $\sum x^2$ should be $149$, not $16$.

Do you want more details or have any questions?

Related Questions:

1. How does summation notation work in general for a given data set?
2. What are the differences between $\sum x$ and $\left(\sum x\right)^2$?
3. How would the results change if the data set included more negative values?
4. How is the calculation of $\sum x^2$ used in variance and standard deviation formulas?
5. Can you explain how $\sum x$ and $\sum x^2$ are used in calculating the mean square?

Tip: Always double-check your calculations to ensure accuracy, especially when working with summation and squared terms.